This pair of chapters discuss Zeno's paradoxes and some of their modern descendants: the ‘dichotomy’, the ‘arrow’, and the ‘supertasks’ of Thompson's lamp and Bernadete. These paradoxes arise from ...
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This pair of chapters discuss Zeno's paradoxes and some of their modern descendants: the ‘dichotomy’, the ‘arrow’, and the ‘supertasks’ of Thompson's lamp and Bernadete. These paradoxes arise from the inifinite divisibility of time and space. For instance, the dichotomy considers dividing a journey into two stages, and then the second stage into half, and the second half of that into half, and so on to infinity: every stage takes a finite time, so shouldn't the whole journey take infinitely long, never to be completed? The problems are challenges to the mathematical description of the world: for instance, the number of metres comprising a journey. The paradoxes reveal confusions in the mathematical nature of infinity, and its application by physics to the world. The chapter explains how a proper understanding of infinity resolves the paradoxes, and demonstrates how these philosophical questions were crucial to the development of mathematical physics.Less

Zeno's Arrow Paradox

Nick Huggett

Published in print: 2010-01-05

This pair of chapters discuss Zeno's paradoxes and some of their modern descendants: the ‘dichotomy’, the ‘arrow’, and the ‘supertasks’ of Thompson's lamp and Bernadete. These paradoxes arise from the inifinite divisibility of time and space. For instance, the dichotomy considers dividing a journey into two stages, and then the second stage into half, and the second half of that into half, and so on to infinity: every stage takes a finite time, so shouldn't the whole journey take infinitely long, never to be completed? The problems are challenges to the mathematical description of the world: for instance, the number of metres comprising a journey. The paradoxes reveal confusions in the mathematical nature of infinity, and its application by physics to the world. The chapter explains how a proper understanding of infinity resolves the paradoxes, and demonstrates how these philosophical questions were crucial to the development of mathematical physics.

This chapter advances the idea that every mathematical argument tells a story by focusing on the biographies of two pioneers: American physicist John Archibald Wheeler and the Bourbaki collective of ...
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This chapter advances the idea that every mathematical argument tells a story by focusing on the biographies of two pioneers: American physicist John Archibald Wheeler and the Bourbaki collective of young French mathematicians. Wheeler viewed mathematical arguments essentially as compound machines; his is a world where instructions pull dimensionality itself out of a Borel sets that he referred to as a “bucket of dust.” Whereas Wheeler's story is a set of linked machine-stories, a hybrid of discovery accounts, speculative machine-like functions and mechanisms, Bourbaki's account is a crystal of symbols. The chapter contrasts Wheeler's way of relating the narrative of mathematics with that of Bourbaki. In particular, it considers Wheeler's machine metaphor and its rejection by Bourbaki; his mathematical physics, and especially his views on gravitational collapse; and Bourbaki's “abstract package.”Less

Structure of Crystal, Bucket of Dust

Peter Galison

Published in print: 2012-03-18

This chapter advances the idea that every mathematical argument tells a story by focusing on the biographies of two pioneers: American physicist John Archibald Wheeler and the Bourbaki collective of young French mathematicians. Wheeler viewed mathematical arguments essentially as compound machines; his is a world where instructions pull dimensionality itself out of a Borel sets that he referred to as a “bucket of dust.” Whereas Wheeler's story is a set of linked machine-stories, a hybrid of discovery accounts, speculative machine-like functions and mechanisms, Bourbaki's account is a crystal of symbols. The chapter contrasts Wheeler's way of relating the narrative of mathematics with that of Bourbaki. In particular, it considers Wheeler's machine metaphor and its rejection by Bourbaki; his mathematical physics, and especially his views on gravitational collapse; and Bourbaki's “abstract package.”

Cambridge University for many years during the Victorian era was one of Europe's foremost training grounds in mathematical physics. From the cosmic sciences of celestial mechanics, thermodynamics, ...
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Cambridge University for many years during the Victorian era was one of Europe's foremost training grounds in mathematical physics. From the cosmic sciences of celestial mechanics, thermodynamics, and electromagnetism to the humbler dynamics of the billiard ball, the boomerang, and the bicycle, the mathematics academics at Cambridge considered themselves the mathematical masters of every known phenomenon of the physical universe. This study aims to deepen the understanding of the nature and historical origins of that expertise by exploring it from the perspective of pedagogy or training. The subject of scientific education has received considerable attention from historians over recent years, yet few studies have made any sustained attempt to use the educational process as a means of investigating scientific knowledge. Little attempt has been made to provide a historiography of the rise of modern mathematical physics in terms of the formation and interaction of communities of trained practitioners. This failure to explore the relationship between learning and knowing is surprising, moreover, as it is now several decades since philosophers of science such as Thomas Kuhn and Michel Foucault drew attention to the importance of training both in the production of knowing individuals and in the formation of the scientific disciplines.Less

Writing a Pedagogical History of Mathematical Physics

Published in print: 2003-07-01

Cambridge University for many years during the Victorian era was one of Europe's foremost training grounds in mathematical physics. From the cosmic sciences of celestial mechanics, thermodynamics, and electromagnetism to the humbler dynamics of the billiard ball, the boomerang, and the bicycle, the mathematics academics at Cambridge considered themselves the mathematical masters of every known phenomenon of the physical universe. This study aims to deepen the understanding of the nature and historical origins of that expertise by exploring it from the perspective of pedagogy or training. The subject of scientific education has received considerable attention from historians over recent years, yet few studies have made any sustained attempt to use the educational process as a means of investigating scientific knowledge. Little attempt has been made to provide a historiography of the rise of modern mathematical physics in terms of the formation and interaction of communities of trained practitioners. This failure to explore the relationship between learning and knowing is surprising, moreover, as it is now several decades since philosophers of science such as Thomas Kuhn and Michel Foucault drew attention to the importance of training both in the production of knowing individuals and in the formation of the scientific disciplines.

This pair of chapters discuss Zeno's paradoxes and some of their modern descendants: the ‘dichotomy’, the ‘arrow’, and the ‘supertasks’ of Thompson's lamp and Bernadete. These paradoxes arise from ...
More

This pair of chapters discuss Zeno's paradoxes and some of their modern descendants: the ‘dichotomy’, the ‘arrow’, and the ‘supertasks’ of Thompson's lamp and Bernadete. These paradoxes arise from the inifinite divisibility of time and space. For instance, the dichotomy considers dividing a journey into two stages, and then the second stage into half, and the second half of that into half, and so on to infinity: every stage takes a finite time, so shouldn't the whole journey take infinitely long, never to be completed? The problems are challenges to the mathematical description of the world: for instance, the number of metres comprising a journey. The paradoxes reveal confusions in the mathematical nature of infinity, and its application by physics to the world. The chapter explains how a proper understanding of infinity resolves the paradoxes, and demonstrates how these philosophical questions were crucial to the development of mathematical physics.Less

Zeno's Paradoxes

Nick Huggett

Published in print: 2010-01-05

This pair of chapters discuss Zeno's paradoxes and some of their modern descendants: the ‘dichotomy’, the ‘arrow’, and the ‘supertasks’ of Thompson's lamp and Bernadete. These paradoxes arise from the inifinite divisibility of time and space. For instance, the dichotomy considers dividing a journey into two stages, and then the second stage into half, and the second half of that into half, and so on to infinity: every stage takes a finite time, so shouldn't the whole journey take infinitely long, never to be completed? The problems are challenges to the mathematical description of the world: for instance, the number of metres comprising a journey. The paradoxes reveal confusions in the mathematical nature of infinity, and its application by physics to the world. The chapter explains how a proper understanding of infinity resolves the paradoxes, and demonstrates how these philosophical questions were crucial to the development of mathematical physics.

There has been a significant increase recently in activities on the interface between applied analysis and probability theory. With the potential of a combined approach to the study of various ...
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There has been a significant increase recently in activities on the interface between applied analysis and probability theory. With the potential of a combined approach to the study of various physical systems in view, this book is a collection of topical survey articles by leading researchers in both fields, working on the mathematical description of growth phenomena in the broadest sense. The main aim of the book is to foster interaction between researchers in probability and analysis, and to inspire joint efforts to attack important physical problems. Mathematical methods discussed in the book comprise large deviation theory, lace expansion, harmonic analysis, multi-scale techniques, and homogenization of partial differential equations. Models based on the physics of individual particles are discussed alongside models based on the continuum description of large collections of particles, and the mathematical theories are used to describe physical phenomena such as droplet formation, Bose–Einstein condensation, Anderson localization, Ostwald ripening, or the formation of the early universe.Less

Analysis and Stochastics of Growth Processes and Interface Models

Published in print: 2008-07-24

There has been a significant increase recently in activities on the interface between applied analysis and probability theory. With the potential of a combined approach to the study of various physical systems in view, this book is a collection of topical survey articles by leading researchers in both fields, working on the mathematical description of growth phenomena in the broadest sense. The main aim of the book is to foster interaction between researchers in probability and analysis, and to inspire joint efforts to attack important physical problems. Mathematical methods discussed in the book comprise large deviation theory, lace expansion, harmonic analysis, multi-scale techniques, and homogenization of partial differential equations. Models based on the physics of individual particles are discussed alongside models based on the continuum description of large collections of particles, and the mathematical theories are used to describe physical phenomena such as droplet formation, Bose–Einstein condensation, Anderson localization, Ostwald ripening, or the formation of the early universe.

When Isaac Newton published the Principia three centuries ago, only a few scholars were capable of understanding his conceptually demanding work. Yet this esoteric knowledge quickly became accessible ...
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When Isaac Newton published the Principia three centuries ago, only a few scholars were capable of understanding his conceptually demanding work. Yet this esoteric knowledge quickly became accessible in the nineteenth and early twentieth centuries when Britain produced many leading mathematical physicists. This book shows how the education of these “masters of theory” led them to transform our understanding of everything from the flight of a boomerang to the structure of the universe. The book focuses on Cambridge University, where many of the best physicists trained. It begins by tracing the dramatic changes in undergraduate education there since the eighteenth century, especially the gradual emergence of the private tutor as the most important teacher of mathematics. Next the book explores the material culture of mathematics instruction, showing how the humble pen and paper so crucial to this study transformed everything from classroom teaching to final examinations. Balancing their intense intellectual work with strenuous physical exercise, the students themselves—known as the “Wranglers”—helped foster the competitive spirit that drove them in the classroom and informed the Victorian ideal of a manly student. Finally, by investigating several historical “cases,” such as the reception of Albert Einstein's special and general theories of relativity, the book shows how the production, transmission, and reception of new knowledge was profoundly shaped by the skills taught to Cambridge undergraduates.Less

Masters of Theory : Cambridge and the Rise of Mathematical Physics

Andrew Warwick

Published in print: 2003-07-01

When Isaac Newton published the Principia three centuries ago, only a few scholars were capable of understanding his conceptually demanding work. Yet this esoteric knowledge quickly became accessible in the nineteenth and early twentieth centuries when Britain produced many leading mathematical physicists. This book shows how the education of these “masters of theory” led them to transform our understanding of everything from the flight of a boomerang to the structure of the universe. The book focuses on Cambridge University, where many of the best physicists trained. It begins by tracing the dramatic changes in undergraduate education there since the eighteenth century, especially the gradual emergence of the private tutor as the most important teacher of mathematics. Next the book explores the material culture of mathematics instruction, showing how the humble pen and paper so crucial to this study transformed everything from classroom teaching to final examinations. Balancing their intense intellectual work with strenuous physical exercise, the students themselves—known as the “Wranglers”—helped foster the competitive spirit that drove them in the classroom and informed the Victorian ideal of a manly student. Finally, by investigating several historical “cases,” such as the reception of Albert Einstein's special and general theories of relativity, the book shows how the production, transmission, and reception of new knowledge was profoundly shaped by the skills taught to Cambridge undergraduates.

This chapter discusses the rise of mathematical physics in revolutionary France at the beginning of the century and the emergence there of new analytical styles of reasoning about nature. Radical ...
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This chapter discusses the rise of mathematical physics in revolutionary France at the beginning of the century and the emergence there of new analytical styles of reasoning about nature. Radical young mathematicians in England such as Charles Babbage and John Herschel looked enviously on at the great strides achieved in revolutionary and Napoleonic France. Undergraduates at the University of Cambridge, they regarded their alma mater as a reactionary backwater—in both political and scientific terms. Following the revolution, French scientific institutions were overturned, as ancient institutions as well as heads toppled to the ground. The chapter then follows efforts to develop new institutions and new ways of training mathematical physicists in England and the German states to midcentury and beyond.Less

A Revolutionary Science

Published in print: 2005-03-01

This chapter discusses the rise of mathematical physics in revolutionary France at the beginning of the century and the emergence there of new analytical styles of reasoning about nature. Radical young mathematicians in England such as Charles Babbage and John Herschel looked enviously on at the great strides achieved in revolutionary and Napoleonic France. Undergraduates at the University of Cambridge, they regarded their alma mater as a reactionary backwater—in both political and scientific terms. Following the revolution, French scientific institutions were overturned, as ancient institutions as well as heads toppled to the ground. The chapter then follows efforts to develop new institutions and new ways of training mathematical physicists in England and the German states to midcentury and beyond.

Contemporary philosophy of mathematics offers us an embarrassment of riches. But anyone familiar with this area will be aware of the need for new approaches that will pay closer attention to ...
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Contemporary philosophy of mathematics offers us an embarrassment of riches. But anyone familiar with this area will be aware of the need for new approaches that will pay closer attention to mathematical practice. This book provides a unified presentation of this new wave of work in philosophy of mathematics. This new approach is innovative in at least two ways. First, it holds that there are important novel characteristics of contemporary mathematics that are just as worthy of philosophical attention as the distinction between constructive and non constructive mathematics at the time of the foundational debates. Secondly, it holds that many topics that escape purely formal logical treatment — such as visualization, explanation, and understanding — can be nonetheless be subjected to philosophical analysis. The book comprises an introduction and eight sections. Each section consists of a short introduction outlining the general topic followed by a related research article. The eight topics selected represent a broad spectrum of contemporary philosophical reflection on different aspects of mathematical practice: visualization, diagrammatic reasoning and representational systems, mathematical explanation, purity of methods, mathematical concepts, philosophical relevance of category theory, philosophical aspects of computer science in mathematics, philosophical impact of recent developments in mathematical physics.Less

The Philosophy of Mathematical Practice

Published in print: 2008-06-19

Contemporary philosophy of mathematics offers us an embarrassment of riches. But anyone familiar with this area will be aware of the need for new approaches that will pay closer attention to mathematical practice. This book provides a unified presentation of this new wave of work in philosophy of mathematics. This new approach is innovative in at least two ways. First, it holds that there are important novel characteristics of contemporary mathematics that are just as worthy of philosophical attention as the distinction between constructive and non constructive mathematics at the time of the foundational debates. Secondly, it holds that many topics that escape purely formal logical treatment — such as visualization, explanation, and understanding — can be nonetheless be subjected to philosophical analysis. The book comprises an introduction and eight sections. Each section consists of a short introduction outlining the general topic followed by a related research article. The eight topics selected represent a broad spectrum of contemporary philosophical reflection on different aspects of mathematical practice: visualization, diagrammatic reasoning and representational systems, mathematical explanation, purity of methods, mathematical concepts, philosophical relevance of category theory, philosophical aspects of computer science in mathematics, philosophical impact of recent developments in mathematical physics.

This chapter presents the reasons advanced against the circulation theory of lift proposed by non-engineers who worked in the British, and particularly the Cambridge, tradition of mathematical ...
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This chapter presents the reasons advanced against the circulation theory of lift proposed by non-engineers who worked in the British, and particularly the Cambridge, tradition of mathematical physics. If the objections were the expressions of a disciplinary standpoint, located at a specific time and place, then perhaps the resistance to the circulatory theory would be explicable as a clash of cultures, institutions, and practices. Such an explanation would not imply any devaluation of the reasons that were advanced against the circulatory theory. It would not be premised on the assumption that these reasons were not the real reasons for the resistance. On the contrary, the intention would be to take the objections against the theory in full seriousness and to probe further into them. The aim of this chapter is to outline a theory that could explain the negative character of the British response to Lanchester's theory.Less

Two Traditions: Mathematical Physics and Technical Mechanics

David Bloor

Published in print: 2011-11-15

This chapter presents the reasons advanced against the circulation theory of lift proposed by non-engineers who worked in the British, and particularly the Cambridge, tradition of mathematical physics. If the objections were the expressions of a disciplinary standpoint, located at a specific time and place, then perhaps the resistance to the circulatory theory would be explicable as a clash of cultures, institutions, and practices. Such an explanation would not imply any devaluation of the reasons that were advanced against the circulatory theory. It would not be premised on the assumption that these reasons were not the real reasons for the resistance. On the contrary, the intention would be to take the objections against the theory in full seriousness and to probe further into them. The aim of this chapter is to outline a theory that could explain the negative character of the British response to Lanchester's theory.

Are things in the real world governed by the mathematical equations of fundamental theories in physics? If we take seriously the practice of fitting facts into equations, the answer should be no. To ...
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Are things in the real world governed by the mathematical equations of fundamental theories in physics? If we take seriously the practice of fitting facts into equations, the answer should be no. To give a mathematical description of a real phenomenon requires bridge principles. However, given the constraints of theory, even these employ highly idealized fictional objects and processes, more akin to artful theatrical distortions than to true descriptions of things in the world.Less

Fitting Facts to Equations

Nancy Cartwright

Published in print: 1983-06-09

Are things in the real world governed by the mathematical equations of fundamental theories in physics? If we take seriously the practice of fitting facts into equations, the answer should be no. To give a mathematical description of a real phenomenon requires bridge principles. However, given the constraints of theory, even these employ highly idealized fictional objects and processes, more akin to artful theatrical distortions than to true descriptions of things in the world.

How did the vast corpus of mathematical innovation of Henri Poincaré (1854–1912) engage the rationale, and impact the fate, of the notion of the ether in physics? Poincaré sought the ‘true relations’ ...
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How did the vast corpus of mathematical innovation of Henri Poincaré (1854–1912) engage the rationale, and impact the fate, of the notion of the ether in physics? Poincaré sought the ‘true relations’ that adhere in the phenomena—relations that persist irrespective of the choice of a metric geometry and a change in physical theory. This chapter traces how Poincaré embedded utterly new geometries and topological intuitions at the heart of pure mathematics, mathematical physics and philosophy. It demonstrates that Poincaré had no ownership of the physicists’ ether concept and that he viewed the ether as neither necessary nor necessarily a hindrance for further advance. Poincaré attended to the profound and subtle needs regarding space and time within physics by creating profound and subtle mathematics to capture the ‘true relations’, of spacetime. Poincaré thereby rendered the physicists’ ether superfluous while also creating mathematical structures for gravitational and quantum phenomena.Less

Poincaré’s Mathematical Creations in Search of the ‘True Relations of Things’

Connemara Doran

Published in print: 2018-09-13

How did the vast corpus of mathematical innovation of Henri Poincaré (1854–1912) engage the rationale, and impact the fate, of the notion of the ether in physics? Poincaré sought the ‘true relations’ that adhere in the phenomena—relations that persist irrespective of the choice of a metric geometry and a change in physical theory. This chapter traces how Poincaré embedded utterly new geometries and topological intuitions at the heart of pure mathematics, mathematical physics and philosophy. It demonstrates that Poincaré had no ownership of the physicists’ ether concept and that he viewed the ether as neither necessary nor necessarily a hindrance for further advance. Poincaré attended to the profound and subtle needs regarding space and time within physics by creating profound and subtle mathematics to capture the ‘true relations’, of spacetime. Poincaré thereby rendered the physicists’ ether superfluous while also creating mathematical structures for gravitational and quantum phenomena.

History, History of Science, Technology, and Medicine, European Early Modern History

As one of the great Renaissance champions of Aristotle, Lefèvre might be expected to be wary of mathematics in natural philosophy. As humanists, Lefèvre and his circle might be expected to avoid ...
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As one of the great Renaissance champions of Aristotle, Lefèvre might be expected to be wary of mathematics in natural philosophy. As humanists, Lefèvre and his circle might be expected to avoid subjects such as the latitude of forms. Yet in certain dialogues Lefèvre offered an explicit rehabilitation of scholastic mathematical physics. This text assumed a metaphysics of making which allowed Lefèvre, Bovelles, and others to think of mathematical objects as relating to physical causes. They provide an account of knowing as making that responded to fifteenth-century concern about the balance of the active and the contemplative lives. These central moves in the making of a mathematical physics are the subject of this chapter.Less

The Mathematical Principles of Natural Philosophy

Richard Oosterhoff

Published in print: 2018-08-02

As one of the great Renaissance champions of Aristotle, Lefèvre might be expected to be wary of mathematics in natural philosophy. As humanists, Lefèvre and his circle might be expected to avoid subjects such as the latitude of forms. Yet in certain dialogues Lefèvre offered an explicit rehabilitation of scholastic mathematical physics. This text assumed a metaphysics of making which allowed Lefèvre, Bovelles, and others to think of mathematical objects as relating to physical causes. They provide an account of knowing as making that responded to fifteenth-century concern about the balance of the active and the contemplative lives. These central moves in the making of a mathematical physics are the subject of this chapter.

This chapter looks at various efforts during the century to explore the unity of nature, starting with the Romantic movement at the beginning of the century and culminating in the development of ...
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This chapter looks at various efforts during the century to explore the unity of nature, starting with the Romantic movement at the beginning of the century and culminating in the development of ether physics at its end. The worlds of natural philosophy at the beginning of the nineteenth century were changing rapidly. There was a close link throughout the nineteenth century between the ways in which physics as a discipline was organized and the ways in which physics organized the world. For early nineteenth-century Romantic philosophers, natural philosophy required a particular kind of individual. Apprehending nature's hidden unities required someone with the innate capacity to look beneath the surface of events and see what others could not. Midcentury experimental natural philosophers such as William Robert Grove suggested that the natural philosopher needed to be someone educated to look beyond the limitations of particular disciplinary preoccupations and see the wider picture of the correlation of forces. By the end of the century, proponents of energy physics argued that only those like them, deeply trained in the complexities of mathematical physics, could see the world as it really was. It needed their grasp to comprehend the subtle workings of the ether. Their understanding of that subtle and universal medium gave them the ability to police the sciences—to adjudicate what was and what was not an acceptable way of looking at the world.Less

The Romance of Nature

Published in print: 2005-03-01

This chapter looks at various efforts during the century to explore the unity of nature, starting with the Romantic movement at the beginning of the century and culminating in the development of ether physics at its end. The worlds of natural philosophy at the beginning of the nineteenth century were changing rapidly. There was a close link throughout the nineteenth century between the ways in which physics as a discipline was organized and the ways in which physics organized the world. For early nineteenth-century Romantic philosophers, natural philosophy required a particular kind of individual. Apprehending nature's hidden unities required someone with the innate capacity to look beneath the surface of events and see what others could not. Midcentury experimental natural philosophers such as William Robert Grove suggested that the natural philosopher needed to be someone educated to look beyond the limitations of particular disciplinary preoccupations and see the wider picture of the correlation of forces. By the end of the century, proponents of energy physics argued that only those like them, deeply trained in the complexities of mathematical physics, could see the world as it really was. It needed their grasp to comprehend the subtle workings of the ether. Their understanding of that subtle and universal medium gave them the ability to police the sciences—to adjudicate what was and what was not an acceptable way of looking at the world.

A pervasive understanding holds that the foundation of calculus-based mathematical physics was laid by Issac Newton in his epochal treatise Philosophiae naturalis principia mathematica of 1687. In ...
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A pervasive understanding holds that the foundation of calculus-based mathematical physics was laid by Issac Newton in his epochal treatise Philosophiae naturalis principia mathematica of 1687. In this historical understanding, what we now call "classical Newtonian mechanics" was born from Newton's work accomplished in his Principia, and was disseminated and digested throughout Enlightenment Europe as a result of the reception of this book. Before Voltaire challenges this understanding by demonstrating the historical gap separating Newton's work in the Principia from the calculus-based mathematical physics that only later became associated with his name. It also shows the important role played by Continental mathematicians, especially in France, in building from Newton's work, but also that of others such as Gottfried Wilhelm von Leibniz and Nicholas Malebranche, the modern science of analytical mechanics. It further distances this history from the direct life and legacy of Newton by demonstrating the important role that the French Académie Royale des Sciences played in creating the institutional crucible from which this new and innovative science was forged. Treating calculus-based mathematical physics as a contingent historical outcome produced through a wide array of intellectual, cultural, social, and political dynamics, this book frees the history of modern mathematical physics from the Enlightenment mythistory of the so-called "Newtonian Revolution." It does so by narrating a fully contingent cultural history of the birth, contests over, and then establishment of analytical mechanics as a foundational French science in the two decades around 1700.Less

Before Voltaire : The French Origins of "Newtonian" Mechanics, 1680-1715

J.B. Shank

Published in print: 2018-06-08

A pervasive understanding holds that the foundation of calculus-based mathematical physics was laid by Issac Newton in his epochal treatise Philosophiae naturalis principia mathematica of 1687. In this historical understanding, what we now call "classical Newtonian mechanics" was born from Newton's work accomplished in his Principia, and was disseminated and digested throughout Enlightenment Europe as a result of the reception of this book. Before Voltaire challenges this understanding by demonstrating the historical gap separating Newton's work in the Principia from the calculus-based mathematical physics that only later became associated with his name. It also shows the important role played by Continental mathematicians, especially in France, in building from Newton's work, but also that of others such as Gottfried Wilhelm von Leibniz and Nicholas Malebranche, the modern science of analytical mechanics. It further distances this history from the direct life and legacy of Newton by demonstrating the important role that the French Académie Royale des Sciences played in creating the institutional crucible from which this new and innovative science was forged. Treating calculus-based mathematical physics as a contingent historical outcome produced through a wide array of intellectual, cultural, social, and political dynamics, this book frees the history of modern mathematical physics from the Enlightenment mythistory of the so-called "Newtonian Revolution." It does so by narrating a fully contingent cultural history of the birth, contests over, and then establishment of analytical mechanics as a foundational French science in the two decades around 1700.

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and ...
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These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and Savilian Professor of Geometry at Oxford. The proceedings contain twenty-nine articles, including three by Fields medallists (Donaldson, Mori and Yau). The articles cover a wide range of topics in geometry and mathematical physics, including the following: Riemannian geometry, geometric analysis, special holonomy, integrable systems, dynamical systems, generalized complex structures, symplectic and Poisson geometry, low-dimensional topology, algebraic geometry, moduli spaces, Higgs bundles, geometric Langlands programme, mirror symmetry and string theory. These volumes will be of interest to researchers and graduate students both in geometry and mathematical physics.Less

Geometry and Physics: Volume II : A Festschrift in honour of Nigel Hitchin

Published in print: 2018-10-25

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and Savilian Professor of Geometry at Oxford. The proceedings contain twenty-nine articles, including three by Fields medallists (Donaldson, Mori and Yau). The articles cover a wide range of topics in geometry and mathematical physics, including the following: Riemannian geometry, geometric analysis, special holonomy, integrable systems, dynamical systems, generalized complex structures, symplectic and Poisson geometry, low-dimensional topology, algebraic geometry, moduli spaces, Higgs bundles, geometric Langlands programme, mirror symmetry and string theory. These volumes will be of interest to researchers and graduate students both in geometry and mathematical physics.

Only a limited number of models of phase transitions and critical phenomena can be solved exactly. These examples nevertheless play important roles in many aspects including the verification of the ...
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Only a limited number of models of phase transitions and critical phenomena can be solved exactly. These examples nevertheless play important roles in many aspects including the verification of the accuracy of approximation theories such as the mean-field theory and renormalization group. Mathematical methods to solve such examples are interesting in their own right and constitute an important subfield of mathematical physics. In particular the exact solution of the two-dimensional Ising model occupies an outstanding status as one of the founding studies of the modern theory of phase transitions and critical phenomena. The present chapter shows simple but typical examples of exact solutions of classical spin systems such as the one-dimensional Ising model with various boundary conditions, the n-vector model, the spherical model, the one-dimensional quantum $XY$ model, and the two-dimensional Ising model. An account on the Yang-Lee theory will also be given as a set of basic rigorous results on phase transitions.Less

Exact solutions and related topics

Hidetoshi NishimoriGerardo Ortiz

Published in print: 2010-12-02

Only a limited number of models of phase transitions and critical phenomena can be solved exactly. These examples nevertheless play important roles in many aspects including the verification of the accuracy of approximation theories such as the mean-field theory and renormalization group. Mathematical methods to solve such examples are interesting in their own right and constitute an important subfield of mathematical physics. In particular the exact solution of the two-dimensional Ising model occupies an outstanding status as one of the founding studies of the modern theory of phase transitions and critical phenomena. The present chapter shows simple but typical examples of exact solutions of classical spin systems such as the one-dimensional Ising model with various boundary conditions, the n-vector model, the spherical model, the one-dimensional quantum $XY$ model, and the two-dimensional Ising model. An account on the Yang-Lee theory will also be given as a set of basic rigorous results on phase transitions.

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and ...
More

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and Savilian Professor of Geometry at Oxford. The proceedings contain twenty-nine articles, including three by Fields medallists (Donaldson, Mori and Yau). The articles cover a wide range of topics in geometry and mathematical physics, including the following: Riemannian geometry, geometric analysis, special holonomy, integrable systems, dynamical systems, generalized complex structures, symplectic and Poisson geometry, low-dimensional topology, algebraic geometry, moduli spaces, Higgs bundles, geometric Langlands programme, mirror symmetry and string theory. These volumes will be of interest to researchers and graduate students both in geometry and mathematical physics.Less

Geometry and Physics: Volume I : A Festschrift in honour of Nigel Hitchin

Published in print: 2018-10-25

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and Savilian Professor of Geometry at Oxford. The proceedings contain twenty-nine articles, including three by Fields medallists (Donaldson, Mori and Yau). The articles cover a wide range of topics in geometry and mathematical physics, including the following: Riemannian geometry, geometric analysis, special holonomy, integrable systems, dynamical systems, generalized complex structures, symplectic and Poisson geometry, low-dimensional topology, algebraic geometry, moduli spaces, Higgs bundles, geometric Langlands programme, mirror symmetry and string theory. These volumes will be of interest to researchers and graduate students both in geometry and mathematical physics.

The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an ...
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The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an extremely useful part of mathematics, knowledge of which is needed by specialists in many other fields. It provides a model for a large number of more recent developments in areas including manifold topology, global analysis, algebraic geometry, Riemannian geometry, and diverse topics in mathematical physics. This text on Riemann surface theory proves the fundamental analytical results on the existence of meromorphic functions and the Uniformisation Theorem. The approach taken emphasises PDE methods, applicable more generally in global analysis. The connection with geometric topology, and in particular the role of the mapping class group, is also explained. To this end, some more sophisticated topics have been included, compared with traditional texts at this level. While the treatment is novel, the roots of the subject in traditional calculus and complex analysis are kept well in mind. Part I sets up the interplay between complex analysis and topology, with the latter treated informally. Part II works as a rapid first course in Riemann surface theory, including elliptic curves. The core of the book is contained in Part III, where the fundamental analytical results are proved.Less

Riemann Surfaces

Simon Donaldson

Published in print: 2011-03-24

The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an extremely useful part of mathematics, knowledge of which is needed by specialists in many other fields. It provides a model for a large number of more recent developments in areas including manifold topology, global analysis, algebraic geometry, Riemannian geometry, and diverse topics in mathematical physics. This text on Riemann surface theory proves the fundamental analytical results on the existence of meromorphic functions and the Uniformisation Theorem. The approach taken emphasises PDE methods, applicable more generally in global analysis. The connection with geometric topology, and in particular the role of the mapping class group, is also explained. To this end, some more sophisticated topics have been included, compared with traditional texts at this level. While the treatment is novel, the roots of the subject in traditional calculus and complex analysis are kept well in mind. Part I sets up the interplay between complex analysis and topology, with the latter treated informally. Part II works as a rapid first course in Riemann surface theory, including elliptic curves. The core of the book is contained in Part III, where the fundamental analytical results are proved.

This chapter examines the long-standing connection between Galilean mathematical science and the phenomenological philosophy that begins with Husserl's 1910 “Philosophy as Rigorous Science” and ...
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This chapter examines the long-standing connection between Galilean mathematical science and the phenomenological philosophy that begins with Husserl's 1910 “Philosophy as Rigorous Science” and extends to the Crisis. In that early text, Husserl took Galileo's work as a model for phenomenology. But this analogy between the two disciplines later became a problem, for by then the scientific ideal embodied in the Galilean style has caused a crisis at the heart of the sciences. The crisis results from mistaking the ideal ontology of mathematical physics, which depends ultimately on life-world experiences, for the totality of objective reality. This chapter then looks at Husserl's notion of the life-world, considering in particular the distinction between objective sciences following the Galilean style and the science of the life-world.Less

Science, History, and Transcendental Subjectivity in Husserl's Crisis

Michael Friedman

Published in print: 2009-12-18

This chapter examines the long-standing connection between Galilean mathematical science and the phenomenological philosophy that begins with Husserl's 1910 “Philosophy as Rigorous Science” and extends to the Crisis. In that early text, Husserl took Galileo's work as a model for phenomenology. But this analogy between the two disciplines later became a problem, for by then the scientific ideal embodied in the Galilean style has caused a crisis at the heart of the sciences. The crisis results from mistaking the ideal ontology of mathematical physics, which depends ultimately on life-world experiences, for the totality of objective reality. This chapter then looks at Husserl's notion of the life-world, considering in particular the distinction between objective sciences following the Galilean style and the science of the life-world.

The book is an inspirational survey of fundamental physics, emphasizing the use of variational principles. Chapter 1 presents introductory ideas, including the principle of least action, vectors and ...
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The book is an inspirational survey of fundamental physics, emphasizing the use of variational principles. Chapter 1 presents introductory ideas, including the principle of least action, vectors and partial differentiation. Chapter 2 covers Newtonian dynamics and the motion of mutually gravitating bodies. Chapter 3 is about electromagnetic fields as described by Maxwell’s equations. Chapter 4 is about special relativity, which unifies space and time into 4-dimensional spacetime. Chapter 5 introduces the mathematics of curved space, leading to Chapter 6 covering general relativity and its remarkable consequences, such as the existence of black holes. Chapters 7 and 8 present quantum mechanics, essential for understanding atomic-scale phenomena. Chapter 9 uses quantum mechanics to explain the fundamental principles of chemistry and solid state physics. Chapter 10 is about thermodynamics, which is built around the concepts of temperature and entropy. Various applications are discussed, including the analysis of black body radiation that led to the quantum revolution. Chapter 11 surveys the atomic nucleus, its properties and applications. Chapter 12 explores particle physics, the Standard Model and the Higgs mechanism, with a short introduction to quantum field theory. Chapter 13 is about the structure and evolution of stars and brings together material from many of the earlier chapters. Chapter 14 on cosmology describes the structure and evolution of the universe as a whole. Finally, Chapter 15 discusses remaining problems at the frontiers of physics, such as the interpretation of quantum mechanics, and the ultimate nature of particles. Some speculative ideas are explored, such as supersymmetry, solitons and string theory.Less

The Physical World : An Inspirational Tour of Fundamental Physics

Nicholas MantonNicholas Mee

Published in print: 2017-04-13

The book is an inspirational survey of fundamental physics, emphasizing the use of variational principles. Chapter 1 presents introductory ideas, including the principle of least action, vectors and partial differentiation. Chapter 2 covers Newtonian dynamics and the motion of mutually gravitating bodies. Chapter 3 is about electromagnetic fields as described by Maxwell’s equations. Chapter 4 is about special relativity, which unifies space and time into 4-dimensional spacetime. Chapter 5 introduces the mathematics of curved space, leading to Chapter 6 covering general relativity and its remarkable consequences, such as the existence of black holes. Chapters 7 and 8 present quantum mechanics, essential for understanding atomic-scale phenomena. Chapter 9 uses quantum mechanics to explain the fundamental principles of chemistry and solid state physics. Chapter 10 is about thermodynamics, which is built around the concepts of temperature and entropy. Various applications are discussed, including the analysis of black body radiation that led to the quantum revolution. Chapter 11 surveys the atomic nucleus, its properties and applications. Chapter 12 explores particle physics, the Standard Model and the Higgs mechanism, with a short introduction to quantum field theory. Chapter 13 is about the structure and evolution of stars and brings together material from many of the earlier chapters. Chapter 14 on cosmology describes the structure and evolution of the universe as a whole. Finally, Chapter 15 discusses remaining problems at the frontiers of physics, such as the interpretation of quantum mechanics, and the ultimate nature of particles. Some speculative ideas are explored, such as supersymmetry, solitons and string theory.